My question is whether there exist an NP-hard problem that has only a caterpillar as input.
By saying only caterpillar as input, I wanted to emphasize that no function (eg: weights on vertices or edges) or specially chosen vertex (or edge) or anything like that is part of the input.
Instance: a caterpillar $T$
There are two similar question in the site: NP-hard problems on trees, NP-hard problems on paths. Note that all answers for the latter problem needs more than one path in input and/or additional structures (edges weights, for instance). In fact, so are most (but not all) answers of the former problem (on trees) as well.
I feel that it is implausible to have an NP-hard problem that take only a path as input. But, it is possible to have an NP-complete problem that take only a caterpillar as input (isn't it?).
Update: I am interested in problems that are naturally defined on caterpillars. True, we cannot define naturality: but we can sense it when we see it.